Solve the following system of linear inequalities by graphing.
3x – y < 2
x + y > 2
No inequalities were presented. You cannot copy from homework and and paste it here; you have to retype the inequalities.
To solve the system of linear inequalities graphically, we will first graph each individual inequality and then find the region where the shaded areas of both inequalities overlap.
Step 1: Graph the inequality 3x - y < 2:
To graph this inequality, we can rewrite it in slope-intercept form:
y > 3x - 2
To graph the equation y = 3x - 2, we need two points. We can choose any values for x and solve for the corresponding y.
Let's choose x = 0:
y = 3(0) - 2
y = -2
So, we have the point (0, -2).
Let's choose x = 1:
y = 3(1) - 2
y = 1
So, we have the point (1, 1).
Plot these two points and draw a dashed line through them.
Step 2: Graph the inequality x + y > 2:
To graph this inequality, rewrite it in slope-intercept form:
y > -x + 2
To graph the equation y = -x + 2, we can choose two arbitrary x values again:
Let's choose x = 0:
y = -0 + 2
y = 2
So, we have the point (0, 2).
Let's choose x = 1:
y = -1 + 2
y = 1
So, we have the point (1, 1).
Plot these two points and draw a dashed line through them.
Step 3: Determine the shaded region:
Now, we need to determine the shaded region that satisfies both inequalities. The shaded region is where the two dashed lines intersect and overlap.
In this case, the shaded region will be above the line representing the inequality y > 3x - 2 and above the line representing the inequality y > -x + 2.
Shade the area above both lines.
Step 4: Finalize the graph:
Finally, label the shaded region as the solution to the system of linear inequalities.
The graph of the system of linear inequalities is complete.
(Note: The exact intersection point of the two lines is not shown in the diagram, but it should be included to determine the region accurately.)
To solve the system of linear inequalities by graphing, we will plot the corresponding graphs and find the region where the shaded areas overlap. This overlapping region represents the solution to the system.
Let's start by graphing the first inequality: 3x - y < 2.
1. Choose a suitable range for x and y. Let's assume -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5.
2. Convert the inequality into an equation to graph. We have 3x - y = 2.
To find the points on this line, we can either find the x and y-intercepts or choose two arbitrary values for x and solve for y.
Let's find the intercepts:
- When x = 0, we have -y = 2, so y = -2. The y-intercept is (0, -2).
- When y = 0, we have 3x = 2, so x = 2/3. The x-intercept is (2/3, 0).
3. Plot the points (0, -2) and (2/3, 0) on the graph.
4. Draw a straight line passing through these two points. This line represents the inequality 3x - y < 2.
Next, let's graph the second inequality: x + y > 2.
1. Convert the inequality into an equation to graph. We have x + y = 2.
Similar to the first inequality, let's find the intercepts:
- When x = 0, we have y = 2. The y-intercept is (0, 2).
- When y = 0, we have x = 2. The x-intercept is (2, 0).
3. Plot the points (0, 2) and (2, 0) on the graph.
4. Draw a straight line passing through these two points. This line represents the inequality x + y > 2.
Now, we have two lines on the graph representing the inequalities.
The solution to the system of linear inequalities is the shaded region where the two lines intersect.
In this case, since both inequalities are strict (not inclusive of the lines themselves), the shaded region will be below the line 3x - y = 2 and above the line x + y = 2.
Therefore, the solution to the system of linear inequalities graphically is the region below the line 3x - y = 2 and above the line x + y = 2.